0.13 The geometry example as a creative trace

The Euclidean-to-Cartesian example can now be read as a structured trace rather than a slogan about abstraction.

Prior presentation.

Geometric objects, incidence relations, constructions, and synthetic propositions.

Combination.

Coordinates provide an interface between geometric objects and algebraic quantities.

Exploration.

Substitution, elimination, parameter variation, and later differential operations traverse consequences enabled by the interface.

Accommodation.

The accepted representational package expands to include coordinate models and algebraic transformations.

Transport.

Synthetic constructions receive coordinate descriptions; geometric interpretation transports selected algebraic results back.

Obstruction.

Coordinate dependence obscures invariant content and motivates later affine, projective, manifold, and coordinate-free formulations.

Tangent probe.

Differentiating the equation of a curve exposes its local admissible directions without reconstructing the entire curve.

This is a retrospective schematic analysis, not a claim that the historical development followed a formal algorithm or a single theory map. It shows the kind of record a synthetic-creativity system should attempt to build: prior language, bridge, new operations, transported results, limitations, and later repairs.

. Do not infer creativity from abstraction alone. An abstraction earns admission when it preserves relevant prior structure, enables independently valuable constructions or predictions, and remains interpretable across more than the example that suggested it.